This part is concerned with variational theory prior to modern quantum mechanics. The exception, saved for Chapter 10, is electromagnetic theory as formulated by Maxwell, which was relativistic before Einstein, and remains as fundamental as it was a century ago, the first example of a Lorentz and gauge covariant field theory. Chapter 1 is a brief survey of the history of variational principles, from Greek philosophers and a religious faith in God as the perfect engineer to a set of mathematical principles that could solve practical problems of optimization and rationalize the laws of dynamics. Chapter 2 traces these ideas in classical mechanics, while Chapter 3 discusses selected topics in applied mathematics concerned with optimization and stationary principles.

In the quantum theory of interacting electrons, a physically correct theory of time dependence should in principle be formulated as a relativistic quantum field theory. The physical model is that of electrons, each characterized by a probability distribution over space-time events xi, ti, that interact indirectly through the quantized electromagnetic field. This theory is simplified for particular applications by neglecting true radiative effects of quantum electrodynamics, and by passing to the limit of large c, the velocity of light in vacuo.

For direct N-electron variational methods, the computational effort increases so rapidly with increasing N that alternative simplified methods must be used for calculations of the electronic structure of large molecules and solids. Especially for calculations of the electronic energy levels of solids (energy-band structure), the methodology of choice is that of independent-electron models, usually in the framework of density functional theory [189, 321, 90]. When restricted to local potentials, as in the local-density approximation (LDA), this is a valid variational theory for any N-electron system. It can readily be applied to heavy atoms by relativistic or semirelativistic modification of the kinetic energy operator in the orbital Kohn–Sham equations [229, 384].

This part is concerned with variational principles underlying field theories. Chapter 10 develops the nonquantized theory of interacting relativistic fields, emphasizing Lorentz and gauge invariant Lagrangian formalism. The theory of a classical nonabelian gauge field is carried to the point of proving gauge invariance and of deriving the local conservation law for field energy and momentum densities.

In 1926, Schrödinger [365] recognized that the variational theory of elliptical differential equations with fixed boundary conditions could produce a discrete eigenvalue spectrum in agreement with the energy levels of Bohr's model of the hydrogen atom. This conceptually startling amalgam of classical ideas of particle and field turned out to be correct. Within a few years, the new wave mechanics almost completely replaced the ad hoc quantization of classical mechanics that characterized the “old” quantum theory initiated by Bohr. Although the matrix mechanics of Heisenberg was soon shown to be logically equivalent, the variational wave theory became the standard basis of methodology in the physics of electrons.

The nonrelativistic Schrödinger theory is readily extended to systems of N interacting electrons. The variational theory of finite N-electron systems (atoms and molecules) is presented here. In this context, several important theorems that follow from the variational formalism are also derived.

Before undertaking the major subject of variational principles in quantum mechanics, the present chapter is intended as a brief introduction to the extension of variational theory to linear dynamical systems and to classical optimization methods. References given above and in the Bibliography will be of interest to the reader who wishes to pursue this subject in fields outside the context of contemporary theoretical physics and chemistry. The specialized subject of optimization of molecular geometries in theoretical chemistry is treated here in some detail.

Linear systems

Any multicomponent system whose dynamical behavior is governed by coupled linear equations can be modelled by an effective Lagrangian, quadratic in the system variables. Hamilton's variational principle is postulated to determine the time behavior of the system. A dynamical model of some system of interest is valid if it satisfies the same system of coupled equations.

This part extends quantum variational theory to continuum states. In particular, variational principles are developed for wave function continuity at specified energy, which is the usual context of scattering theory. Chapter 7, concerned with multiple scattering theory, lies somewhere between the theory of bound states and true scattering theory. Formalism appropriate to the latter is adapted to computing the electronic structure of large molecules and periodic solids, whose energy levels are determined by consistency conditions for wave function continuity. A variational formalism is derived for energy linearization. Chapter 8 develops variational principles and methods suitable for the true continuum problem of electron scattering at specified energy. Chapter 9 presents methodology, some very recent, that allows rotational and vibrational effects in electron–molecule scattering to be treated as a practicable extension of fixed-nuclei variational theory.

This part introduces variational principles relevant to the quantum mechanics of bound stationary states. Chapter 4 covers well-known variational theory that underlies modern computational methodology for electronic states of atoms and molecules. Extension to condensed matter is deferred until Part III, since continuum theory is part of the formal basis of the multiple scattering theory that has been developed for applications in this subfield. Chapter 5 develops the variational theory that underlies independent-electron models, now widely used to transcend the practical limitations of direct variational methods for large systems. This is extended in Chapter 6 to time-dependent variational theory in the context of independent-electron models, including linear-response theory and its relationship to excitation energies.

The idea that laws of nature should satisfy a principle of simplicity goes back at least to the Greek philosophers [436]. The anthropomorphic concept that the engineering skill of a supreme creator should result in rules of least effort or of most efficient use of resources leads directly to principles characterized by mathematical extrema. For example, Aristotle (De Caelo) concluded that planetary orbits must be perfect circles, because geometrical perfection is embodied in these curves: “… of lines that return upon themselves the line which bounds the circle is the shortest. That movement is swiftest which follows the shortest line”. Hero of Alexandria (Catoptrics) proved perhaps the first scientific minimum principle, showing that the path of a reflected ray of light is shortest if the angles of incidence and reflection are equal.

As theoretical physics and chemistry have developed since the great quantum revolution of the 1920s, there has been an explosive speciation of subfields, perhaps comparable to the late Precambrian period in biological evolution. The result is that these life-forms not only fail to interbreed, but can fail to find common ground even when placed in proximity on a university campus. And yet, the underlying intellectual DNA remains remarkably similar, in analogy to the findings of recent research in biology. The purpose of this present text is to identify common strands in the substrate of variational theory and to express them in a form that is intelligible to participants in these subfields. The goal is to make hard-won insights from each line of development accessible to others, across the barriers that separate these specialized intellectual niches.

Another great revolution was initiated in the last midcentury, with the introduction of digital computers. In many subfields, there has been a fundamental change in the attitude of practicing theoreticians toward their theory, primarily a change of practical goals. There is no longer a well-defined barrier between theory for the sake of understanding and theory for the sake of predicting quantitative data. Given modern resources of computational power and the coevolving development of efficient algorithms and widely accessible computer program tools, a formal theoretical insight can often be exploited very rapidly, and verified by quantitative implications for experiment. A growing archive records experimental controversies that have been resolved by quantitative computational theory.

In quantum electrodynamics (QED), the classical electromagnetic field Aμ of Maxwell and the electronic field ψ of Dirac are given algebraic properties (Bose–Einstein and Fermi–Dirac quantization, respectively), and through their interaction account for almost all physical phenomena that can be observed in ordinary human circumstances. The relativistic theory is derived from Hamilton's principle for an action defined by the space-time integral of a Lorentz invariant Lagrangian density [373]. This same action integral can be used to develop the diagrammatic perturbation theory of Feynman [121]. The cited references describe the formalism and methodology which demonstrate that QED is in remarkable agreement with all empirical data to which it is applicable. Classical and quantized QED will be used here to introduce the basic formalism of field theory, including the variational theory of invariance properties. This theory, especially gauge invariance, is central to recent developments of electroweak theory (EWT) and quantum chromodynamics (QCD).

Electron–molecule scattering data, observed experimentally or computed with methodology available as of 1980, was reviewed in detail by Lane [215]. If there were no nuclear motion, electron–molecule scattering would differ from electron–atom scattering only because of the loss of spherical symmetry and because of the presence of multiple Coulomb potentials due to the atomic nuclei. This is already a formidable challenge to theory, exemplified by the qualitative increase in computational difficulty and complexity between atomic theory and molecular theory for electronic bound states. While bound-state molecular computational methods have been extended to fixed-nuclei electron scattering [49, 178], an effective and computationally practicable treatment of rovibrational (rotational and vibrational) excitation requires a significant and historically challenging extension of bound-state theory.

Despite the simple and universal structure of the nonrelativistic Hamiltonian for N interacting electrons, it produces a broad spectrum of physical and chemical phenomena that are difficult to conceptualize within the full N-electron theory. Starting with the work of Hartree [162] in the early years of quantum mechanics, it was found to be very rewarding to develop a model of electrons that interact only indirectly with each other, through a self-consistent mean field. A deeper motivation lies in the fact that the relativistic quantum field theory of electrons is explicitly described by a field operator that corresponds more closely to a oneparticle model wave function than to that of the Schrödinger N-electron theory. The fundamental characterization of this electron field by Fermi–Dirac statistics is directly applicable to the mean-field theory, using concepts of statistical occupation numbers determined by effective one-electron orbital energy values. The variational theory appropriate to such independent-electron models is developed in this chapter.

Long ago, Paul Lévy invented a strange family of random walks – where each segment has a very broad probability distribution. These flights, when they are observed on a macroscopic scale, do not follow the standard Gaussian statistics. When I was a student, Lévy's idea appeared to me as (a) amusing, (b) simple – all the statistics can be handled via Fourier transforms – and (c) somewhat baroque: where would it apply?

As often happens with new mathematical ideas, the fruits came later. For example, é. Bouchaud proved that adsorbed polymer chains often behave like Lévy flights. In a very different sector, J.P. Bouchaud showed the role of Lévy distributions in risk evaluation. Now we meet a third major example, which is described in this book: cold atoms.

The starting point is a jewel of quantum physics: we think of an atom in a state of 0 translational momentum p = 0 (zero Doppler effect), inside a suitably prescribed laser field. For instance, with an angular momentum J = 1 we can have two ground states │+〉 and │−〉, and one excited state │0〉. The particular state │+〉+│−〉 has an admirable property: it is entirely decoupled from the radiation and can live for an indefinitely long time. It is thus possible to create a trap (around p = 0 in momentum space) in which the atoms will live for very long times: this so-called ‘ subrecoil laser cooling’ has been a major advance of recent years.